Download Quantum computers

aster, smaller, less
expensive—that’s
what we want our
computers to be.
No matter how
good a computer
chip is…it’s never good enough.
We all know, at least to some extent,
how today’s computer works. The digital logic behind it is actually very simple and fairly intuitive: an array of 0s
and 1s represents a number. One that is
easy to store, manipulate and to handle
with our present level of technology.
Today, this digital logic is implemented using transistors: devices that
are widely believed to have been the
greatest invention of the 20th century.
Much time and money is being invested in developing new, smaller ones.
(Just visit intel.com’s news section.)
However, what is being done is
merely a change of technology or a
change of techniques: adjustments in
the manufacturing or design procedure
of semiconductors that may result in
better transistors, hence, better chips.
(The other way is to develop better
techniques for using these transistors,
to lay them out in such a way that better circuits are built.)
An alternative to this approach to
advancing technology would be to
think about the basic philosophy behind
our computers: the digital logic they are
founded on.
A simple run through suggests an
interesting solution: as we want our
computers to get smaller, we eventually
would want the smallest unit of these
computers, the bit, to be the smallest
possible unit of matter: the atom.
However, using atoms as digital bits
will start a completely new era in computer design. Atoms cannot be simply
manipulated and used like the bits built
with transistors. The behavior of matter
on the atomic scale follows the rules of
modern physics. This behavior cannot
be understood in terms of our classical
description of the world (i. e. Newtonian
Mechanics or Maxwell’s Equations in
Electromagnetics).
The physical theory dealing with
such behavior is called quantum
mechanics. Its use in the computer
industry will most probably cause a
revolution in the way we use and
understand computers. We are going to
describe how such a quantum
computer—a computer based on the
rules of quantum mechanics—may
work, and how it is going to give us
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incredible speed and problem-solving
power that we only imagine now from
watching Star Trek.
What is quantum mechanics?
The purpose of physics, or more generally the purpose of science, is to generate a model of the real world around
us. Physical theories are, therefore,
models with which we try to understand
or explain why and how various phenomena around us happen. A good theory is also capable of making correct
predictions, i. e. describing never-done
experiments and telling you how their
results will turn out. We then can perform tests in the laboratory to see
whether or not these predictions hold.
Quantum
computers
Pedram Khalili
Amiri
redefining logic
Quantum mechanics is just one of
these physical theories. It was developed to explain some of the experiments’ results in the first half of the
20th century. We don’t need to concern
ourselves about the complete theory;
we can just touch on the main points in
order to develop our discussion about
quantum computers.
One good way to quickly grasp the
basics of quantum mechanics is to consider the well-known problem of light's
duality. Light was initially believed to
be a build up of particles. Newton, for
example, was one of the most important scientists who tried to explain
light’s behavior this way.
Then, a Scottish physicist, James C.
Maxwell, showed theoretically that
light is a combination of alternating
electrical and magnetic fields, hence a
wave. Experiments were done and the
0278-6648/02/$17.00 © 2002 IEEE
theory proved to be valid. The famous
experiment done by Heinrich Hertz,
was just one of the many successful
experiments. It gave evidence for the
propagation of electromagnetic waves
using a simple transmitter-detector system. (Hertz actually didn’t use lightwaves, but his experiment gave a strong
proof of the validity of Maxwell’s theory in general.)
Later on, new experiments were
done which again forced scientists to
believe in the particle nature of light.
Einstein, for example, explained the
photoelectric effect in terms of the particle nature of light, and obtained a
noble prize for it. So is light made up of
particles or is it a wave?
The answer is very beautifully
contained in the words of Richard
Feynman—one of the greatest
physicists of the 20th century— "it
is like neither."
Light doesn’t behave exactly like
a wave; it also doesn’t behave
exactly like a particle—at least, not
always. So it must be something
else, something in between, or some
kind of combination of the
two…that's what we may think. But
let's look at the answer.
The more complete description
of light—and of any other "thing" in
this world, including matter—states:
It is made up of particles, but these
particles are distributed wave-like,
hence we see both effects. To
be a little more technical, the
probability of finding these
particles in a certain point (of
space and time) has a wavelike distribution.
But that rough description is obviously not enough. We need a quantitative way, an equation, to solve for the
probability of finding a particle in a
special point (of space and time). This
is implemented by the Schrödinger
Equation, named after Erwin
Schrödinger, who first formulated it.
This differential equation is basically
the principle of conservation of energy,
translated to the new probability-wave
formalism of quantum mechanics.
Using this equation, we can find the
probability function for any particle in
any given set of environmental and initial conditions.
We don’t need to deal with the
whole mathematics of the problem, as it
is not essential to our brief discussion
of quantum computing. We will only
use some of its results.
IEEE POTENTIALS
Uncertainty and quantum states
The first thing we think of when confronted with probabilities is uncertainty.
In fact, probabilities always tell us something, but never everything about a phenomenon. When we say that the probability of seeing ‘1’ in rolling a die is 1/6,
we mean that on average, we will see a
"1" in every six throws. It
doesn’t tell us what we will
see in one throw, nor does it
guarantee that we will see,
for example, one hundred 1s
in 600 tosses.
You
may
have
heard about Heisenberg’s
Uncertainty principle. This
principle is a little more complicated, and different, from
the dice example. But that
example can help us understand why we can’t determine the position of a particle
exactly (i.e. without error).
Heisenberg’s Uncertainty
principle essentially says
that position doesn’t really
have a meaning for something like an electron (provided that we know at least
"something" about its velocity, or momentum), as the
error in measuring its position is too great. For ordinary objects, like a chair or a
stone, this minimum error is
too small to be accounted
for; but in analyzing the
behavior of an electron, it
plays a crucial role.
So, how can we describe
the circumstances an electron is in if we don't use
concepts like position, speed
and such? The solution is
provided by Schrödinger's
equation itself.
The new term we use to describe the
behavior of electrons, or any other thing
that is small enough to show quantum
effects, is called a quantum state. In
fact, each quantum state is associated
with a probability distribution for the
position of the particle, not with the
position itself. It is also associated with
some restrictions on some of the particle’s properties, such as energy or spin.
This simply means that to analyze the
behavior of a particle such as an electron, photon, etcetera, we will have to
use the Schrödinger equation. This will
lead us to a series of so called quantum
states. In each of those quantum states,
Inset: When a magnetic cobalt atom is placed at a focus point of an elliptical corral
(upper left), some of its properties also appear at the other focus (lower left), where no
atoms exist. When the cobalt atom is not placed at a focus point (upper right), the
mirage disappears (lower right). In the big picture, a change in the surface electrons
due to the cobalt’s magnetism—the Kondo resonance—appears as a bright spot at
each focus. This phenomenon, called the “quantum mirage” effect, uses the wave
nature of electrons rather than a wire. (PHOTO CREDITS: IBM)
DECEMBER 2002/JANUARY 2003
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the electron has a well-defined probability distribution, and energy level,
or/and spin state and so on.
These general concepts of quantum
mechanics provide the foundation to discuss Quantum Information Theory, a possible way to build computers in the future.
ment will destroy the actual state of the
system. It will collapse to one of the
two states depending on the coefficients, a and b. In fact, the probability of
finding |0> or |1> is:
Quantum bits
Since the result has to be one of
those two states, we have:
Let’s consider a quantum system (a
particle in specifically defined environmental conditions) with which we can
associate two quantum states. This can
be a particle having the spin states "up"
or "down," for example. We will denote
the first one of these two states with |0>,
and the other one with |1>, similar to
the 0 and 1 states of the classical theory
of computation.
But there are two differences when
compared with the classical theory:
1) The |0> and |1> states are quantum states. Thus, they do not denote for
example the voltage of a pin on some
chip, but the state associated with one of
the quantized properties of some quantum system. This property can be the
spin, or energy level, or angular
momentum, etc. of an electron in some
potential field.
2) Most importantly, these quantum
states do not have the intuitive properties of their classical counterparts. A
great difference when compared with
classical bits—in fact, this difference
has in it the heart of quantum computation—is that a quantum system does not
necessarily have to be in one of the two
states |0> and |1>. It can simply be in
any superposition of the two states.
Thus, we can write:
state = a |0> + b |1> .
This is a very straightforward result
deduced from the Schrödinger equation.
But we don’t care about the mathematics
here. The interpretation of a superposed
state would involve the uncertainty principle: we cannot say definitely which
state the system is in. The point is that
we can only measure one of the two
states |0> and |1> when carrying out a
measurement operation on the system.
This means that, for a system which is
in a state of the form just described, a measurement will only yield |0> or |1>, not the
actual state before the measurement.
Further, according to a form of the
uncertainty principle, the actual state of
the system after the measurement will
be the measured state: |0> or |1>. This
means that carrying out the measure-
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P( |0> ) = a2
and
P( |1> ) = b2
a2 + b2 = 1
Actually, the mathematics to really
show that all these results are true
would be rather complicated, so we
won’t give them here.
To sum up, a system with the properties mentioned is called a quantum bit, or
qubit. A qubit can be in a superposition
of its two basic states, and a measurement on it will yield only one (and exactly one) of these two states. Thus, the
effect of the other one (the coefficient a
or b associated with it) will disappear.
Obviously, building a computer
based on qubits means we have to dramatically change our approach. The
computer would have to work in a completely new way. The algorithms needed to manipulate and make qubits function are called quantum algorithms.
Quantum algorithms
We will now briefly discuss how a
qubit can be manipulated to realize the
counterparts of algorithms and programs—as we know them now—on a
quantum computer. You may have
already guessed that what we actually
said is that any state of a qubit can be
represented with a vector-like form.
That is, we can write:
state = a |0> + b |1>
where:
a2 + b2 = 1 .
Therefore, all we have to do is carry
out some kind of transformation on the
vector (actually on the qubit represented
by the vector). The transformation
should increase the possibility of the
appearance of the correct answer ( |0>
or |1> ) depending on what the calculation is and what answer is correct for
that calculation. Keep in mind, that this
qubit is just one in a series of qubits carrying some information, a so-called
quantum register. A series of such
transforms carried out on a qubit is then
called a quantum algorithm.
You may ask, however, how these
transformations can be carried out with-
out destroying the qubit’s state (since
we said that measuring a qubit will
destroy its content and make it collapse
into one state or another). The answer is
simple. These transformations do not
have anything to do with measurement.
Instead, they make use of the interference property of the probability waves
of the qubits.
We know that waves—any type—
have an interference property. This
means that they can make each other
disappear (interfere destructively) or
emphasize each other (interfere constructively). That’s just the same for
probability waves. Without getting too
specific, the transformations on qubits
are actually actions taken to make
unwanted (incorrect) answers to interfere destructively, leaving only the correct answer with a considerable probability of appearance.
Several quantum algorithms have
already been designed and, even tested,
on tiny quantum computers with a small
set of qubits. The results have been
tremendous: quantum algorithms show
the potential of being much faster than
their classical counterparts.
A good example is Shor’s algorithm
for factoring integer numbers. Actually,
the ability of classical computers to factor
numbers is so restricted that it is widely
relied upon to develop secure codes. In
contrast, quantum computers using
Shor’s algorithm would be fast enough to
make almost any security system, based
on this principle, unsafe. That is certainly
one of the main reasons why military
research institutions have been attracted
by the field and are funding research programs to discover its potential.
Ironically, quantum computation theory itself delivers a unique, completely
safe way of encoding data. Only the
person who is intended to receive the
message will actually receive it, other
people trying to figure out the code
(transformations needed to extract the
message) will get only one try. This is
based on the uncertainty principle and
the collapsing of quantum bits upon
measurement. If the first try is not a hit
(and it usually isn’t), they will have
destroyed the message, they will not be
able to try any alternatives.
Another one is the Grover's
Algorithm. This is an algorithm for
searching through lists, to find someone’s phone-number for example.
Grover’s algorithm is also considerably
faster than any classical one, especially
at large numbers of entries in the data-
IEEE POTENTIALS
Entanglement and teleportation
One of the strangest predictions of the
quantum theory is a phenomenon that is
called entanglement. This phenomenon
is closely related to a technique called
teleportation. A discussion of quantum
computation theory cannot be regarded
as complete unless these two terms are
defined and explained somewhat.
To put it simply, qubits can be linked
in such a way as to share the same destiny. That seems mysterious, actually
what it says is mysterious, and the phenomenon is called entanglement. If two
quantum systems (i.e. qubits) are entangled, and measuring one of the two has
given the result, say |0>, measuring the
other one will only give |1>.
Regardless how far the two systems
are from each other, this effect occurs
instantaneously. That is, qubits that are
entangled will feel the effect of measurements carried out on the other one
without any delay, even if the distance
between them is several light-years.
Note: this does not mean a faster-thanlight communication, but as said before,
it is better interpreted as the two systems sharing the same destiny.
This has been experimentally proven
by the famous EPR experiment, and the
entangled pairs are commonly called
EPR pairs. The experiment is named
after Einstein, Podolsky and Rosen who
actually devised it as a way to show that
quantum theory is not consistent with
reality. It consisted of examining
whether entangled pairs really feel the
effect of measurements made on each
other instantly or not. Einstein, Podolsky
and Rosen predicted this to be impossible, and hence wanted to use it as a
proof of the incompleteness of quantum
theory. Contrary to their prediction, the
EPR pairs behaved exactly in the way
quantum theory had described.
Based on the concept of entanglement, a way of "transporting" (or to be
more accurate, "reproducing at another
DECEMBER 2002/JANUARY 2003
place the state of") qubits has been
found that is called teleportation. This
has also been carried out experimentally
on very simple systems.
Teleportation is being considered as
a way of transmitting the contents of
quantum registers over a distance. This
is an essential issue to resolve if any
practical quantum computer is to be
built in the future.
Decoherence of qubits &
possible practical implementations
In discussing qubits and quantum
algorithms, we did not mention one very
important thing: To do all that can be
done with a qubit, there is a very limited
timeframe to work in. That is because
superpositions of states (states of the
form: a |0> + b |1> ) are generally very
unstable, and will collapse into one of the
pure states |0> or |1> quickly as a result
of interactions with the environment.
This result is also easily derived from
the solution of the Schrödinger equation,
but we are not going to prove it here.
The time remaining before the state of a
qubit is completely destroyed is called
the decoherence time. The whole
process is known as decoherence.
The decoherence time is an extremely important factor when considering
practical implementations of quantum
computers. To build a quantum computer that works, we will have to design the
system in such a way that environmental effects are minimized as much as
possible so as to increase the decoherence time. A reasonable amount of calculations should be carried out on a
qubit before it decoheres.
Ways of building such systems have
and are still being investigated. Some of
the most promising ones are systems
based on Nuclear Magnetic Resonance
(NMR) and the so-called Ion Traps
(systems in which ions are cooled down
to such a low energy that their vibrational states can be used as qubits).
In addition, error correction algorithms are being investigated for
qubits to be recovered if their states
were affected by, say, transmission
over a distance. The function of these
algorithms is essentially the same as
that of the error correction algorithms
of today’s digital communication systems. Although, their structure is completely different from the classical
ones used.
Read more about it
•
M.A.Nielsen,
I.A.Chuang,
Quantum Computation and Quantum
Information, Cambridge University
Press, 2000.
• C. Paquin, "Computing in the
Quantum World," [online], Available:
http://www.iro.umontreal.ca/~paquin/Q
u/quantumComp.pdf
• J. Mullins, "The Topsy Turvy
World of Quantum Computing," IEEE
Spectrum, vol. 38, no. 2, pp. 42-49,
February 2001.
• J. Preskill, "Quantum Computing:
Pro and Con," Proc. Roy. Soc. Lond.
A454, pp. 469-486, 1998.
• D. Deutsch, "Quantum Theory, the
Church-Turing principle and the universal quantum computer," Proc. Roy. Soc.
Lond. A400, 96, 1985.
• R. P. Feynman, R. B. Leighton, M.
Sands, The Feynman Lectures on
Physics,Volume III, Addison-Wesley,
1966.
• S. Gasiorowicz, Quantum Physics,
Second Edition, John Wiley & Sons,
1996.
• In addition, lots of useful information on the subject can be found at the
website of the Oxford University Center
for
Quantum
Computation:
<http://www.qubit.org/>
About the author
Pedram Khalili Amiri was born in
Tehran, Iran. Currently, he is an undergraduate student at Sharif University of
Technology in Tehran, where he is
studying Electrical Engineering. His
interests
include
Quantum
Computation, Semiconductor Device
Physics and Fabrication, and
Microelectronic Circuits. He would like
to pursue an MS degree in either solidstate electronics or applied physics. He
is also an experienced web-developer
and has worked in the field for a company in Tehran.
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PHOTO CREDIT: IBM
base. In a database with N entries, a classical search algorithm would normally
need N/2 tries for finding the desired
item, where as Grover’s algorithm needs
only a number on the order of the squareroot of N, which is much faster.
It would not be appropriate to
develop the concept of quantum algorithms further in this introductory article, for a discussion of this subject see
the book by Nielsen and Chuang and
the qubit.org website (See Read more
about it).